Problem 42 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

Step-by-Step Solution

Answer

The statement makes sense because using both the cross-products principle and multiplying both sides by the least common denominator correctly solves the equation.

Step 1: Using the Cross-Products Principle

The cross-products principle states that, in an equation of the form \(\frac{a}{b} = \frac{c}{d}\), ad should equal bc. Here, it means that \(x*6\) should equal \(9*4\). If we solve for x, we get \(x = \frac{9*4}{6}\).

Step 2: Using the Least Common Denominator

If we multiply both sides by 18, the least common denominator of 9 and 6, the equation \(\frac{x}{9} = \frac{4}{6}\) becomes \(2x = 12\). Solve for x by dividing each side by 2, we have \(x = \frac{12}{2}\).

Step 3: Comparison

Comparing the results, we see that using both methods gives the same result for x. So, the statement 'I can solve \(\frac{x}{9} = \frac{4}{6}\) by using the cross-products principle or by multiplying both sides by 18, the least common denominator.' makes sense because both methods accurately solve the equation.